Nevermind, that way should work. However, you seem to have miscalculated your h_n. One trick you can do (if you have time and the ability to do so) is to plot the series (to a reasonable amount of terms) and see if it matches the function you're trying to represent.

Anyway, you're actually on your way. Continue for the "nth" value of n. Basically you take away all the summations and you can solve the second-order ODE for a function of t. You should be able to find the n's in the x function by using the boundary conditions and for the n's in the time function by using the initial conditions.

Nevermind, that way should work. However, you seem to have miscalculated your h_n. One trick you can do (if you have time and the ability to do so) is to plot the series (to a reasonable amount of terms) and see if it matches the function you're trying to represent.

Anyway, you're actually on your way. Continue for the "nth" value of n. Basically you take away all the summations and you can solve the second-order ODE for a function of t. You should be able to find the n's in the x function by using the boundary conditions and for the n's in the time function by using the initial conditions.

That can't be right because I used WolframAlpha to compute the integral.